A Wasserstein distance does more than assign a transport cost. It equips a space of probability measures with a metric that simultaneously records weak convergence and control of tails. The geometry developed later rests on this topology.
Definition and finiteness
Definition. Let \((X,d)\) be Polish and \(1\le p<\infty\). For \(\mu,\nu\in\mathcal P_p(X)\), define \[ W_p(\mu,\nu) =\left( \inf_{\pi\in\Pi(\mu,\nu)} \int_{X\times X}d(x,y)^p\,\mathrm{d}\pi(x,y) \right)^{1/p}. \] An optimizer exists because \(d^p\) is continuous and nonnegative.
Finiteness. For any base point \(x_0\), \[ W_p(\mu,\nu)^p \le 2^{p-1}\left( \int d(x,x_0)^p\,\mathrm{d}\mu +\int d(y,x_0)^p\,\mathrm{d}\nu \right). \]
Proof. Use the product coupling \(\mu\otimes\nu\) and the inequality \(d(x,y)^p\le2^{p-1}\{d(x,x_0)^p+d(y,x_0)^p\}\).
The metric axioms
Theorem. \(W_p\) is a metric on \(\mathcal P_p(X)\).
Proof. Nonnegativity and symmetry are immediate. If \(W_p(\mu,\nu)=0\), an optimal plan \(\pi\) satisfies \(d(x,y)=0\) almost surely, so it is concentrated on the diagonal and its two marginals coincide.
For the triangle inequality, choose \(\pi_{12}\in\Pi(\mu_1,\mu_2)\) and \(\pi_{23}\in\Pi(\mu_2,\mu_3)\), and glue them to \(\gamma\) on \(X^3\). Its \((1,3)\)-marginal is a coupling of \(\mu_1,\mu_3\). Minkowski's inequality gives \[ \begin{aligned} W_p(\mu_1,\mu_3) &\le\left(\int d(x_1,x_3)^p\,\mathrm{d}\gamma\right)^{1/p}\\ &\le\left(\int d(x_1,x_2)^p\,\mathrm{d}\gamma\right)^{1/p} +\left(\int d(x_2,x_3)^p\,\mathrm{d}\gamma\right)^{1/p}. \end{aligned} \] Taking infima over the two plans proves \(W_p(\mu_1,\mu_3)\le W_p(\mu_1,\mu_2)+W_p(\mu_2,\mu_3)\).
Elementary comparisons
Proposition. If \(1\le p\le q\) and \(\mu,\nu\in\mathcal P_q(X)\), then \[ W_p(\mu,\nu)\le W_q(\mu,\nu). \] Also, \[ W_p(\mu,\delta_{x_0})^p=\int d(x,x_0)^p\,\mathrm{d}\mu(x). \]
Proof. For every coupling, the \(L^p\)-norm of \(d(x,y)\) is at most its \(L^q\)-norm; take infima. A coupling with \(\delta_{x_0}\) is unique, namely \(\mu(\mathrm{d}x)\delta_{x_0}(\mathrm{d}y)\), giving the second formula.
On a space of finite diameter \(D\), interpolation gives \(W_q^q\le D^{q-p}W_p^p\). On unbounded spaces no reverse comparison is possible without stronger moment bounds.
Characterization of \(W_p\)-convergence
Topology theorem. For \(\mu_n,\mu\in\mathcal P_p(X)\), the following are equivalent:
(i) \(W_p(\mu_n,\mu)\to0\).
(ii) \(\mu_n\Rightarrow\mu\) and, for one base point \(x_0\), \[ \int d(x,x_0)^p\,\mathrm{d}\mu_n(x) \to \int d(x,x_0)^p\,\mathrm{d}\mu(x). \]
(iii) \(\mu_n\Rightarrow\mu\) and the \(p\)-th moments are uniformly integrable: \[ \lim_{R\to\infty}\sup_n \int_{\{d(x,x_0)>R\}}d(x,x_0)^p\,\mathrm{d}\mu_n(x)=0. \]
Proof. If \(W_p(\mu_n,\mu)\to0\), take nearly optimal couplings \((X_n,X)\). Then \(d(X_n,X)\to0\) in \(L^p\), hence in probability, which implies weak convergence. The reverse triangle inequality \[ |d(X_n,x_0)-d(X,x_0)|\le d(X_n,X) \] and the \(L^p\) norm inequality give convergence of \(p\)-th moment norms and uniform integrability.
Conversely, weak convergence admits couplings with \(X_n\to X\) almost surely by the Skorokhod representation theorem. Condition (iii) makes \(d(X_n,X)^p\) uniformly integrable because it is bounded by \(2^{p-1}\{d(X_n,x_0)^p+d(X,x_0)^p\}\). Vitali's theorem yields \(\mathbb E d(X_n,X)^p\to0\), hence \(W_p(\mu_n,\mu)\to0\). Equivalence of (ii) and (iii) follows by truncating the moment function and using weak convergence on bounded continuous truncations.
Completeness and separability
Theorem. If \((X,d)\) is complete and separable, then \((\mathcal P_p(X),W_p)\) is complete and separable.
Proof. For completeness, take a \(W_p\)-Cauchy sequence and extract a subsequence with \(W_p(\mu_{n_k},\mu_{n_{k+1}})\le2^{-k}\). Glue nearly optimal successive couplings to random variables \(X_k\) on one space. Then \[ \sum_k\|d(X_k,X_{k+1})\|_{L^p}<\infty. \] Hence \(X_k\) is almost surely Cauchy along a further argument and converges to an \(X\)-valued variable \(X_\infty\), using completeness of \(X\). Minkowski and tail summation give \(\|d(X_k,X_\infty)\|_{L^p}\to0\), so \(W_p(\mu_{n_k},\mathcal L(X_\infty))\to0\). The Cauchy property brings the full sequence to the same limit.
For separability, choose a countable dense subset \(D\subset X\). Finitely supported probability measures on \(D\) with rational weights form a countable family. Truncation to a compact high-mass set, quantization by nearby points of \(D\), and rational approximation of weights show that this family is \(W_p\)-dense.
Relative compactness
Compactness criterion. A family \(\mathcal K\subset\mathcal P_p(X)\) is relatively compact in \(W_p\) if it is tight and its \(p\)-th moments are uniformly integrable. On a proper metric space, a uniform bound on a moment of order \(q>p\) implies these conditions.
The extra tail condition is essential. Weak compactness alone permits a small amount of mass to escape to infinity while carrying nonvanishing \(p\)-transport cost, as the example on page 3 showed.
The \(W_\infty\) distance
For bounded-support measures one may also define \[ W_\infty(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)} \operatorname*{ess\,sup}_{(x,y)\sim\pi}d(x,y). \] It minimizes the largest displacement rather than an average displacement. It is the limit of \(W_p\) as \(p\to\infty\) under suitable boundedness assumptions, but its topology and geodesics can be more rigid.